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Calculus : What is it?
Calculus - What is it ? Calculus simply deals with '''Rates Of Change'''. It provides a way for us to construct relatively simple models of change, and to deduce their consequences. Calculus approaches the paths of objects in motion as curves, or functions, and then determines the value of these functions to calculate their rate of change, area, or volume. Here is a basic example of how calculus differs from the Regular Math we have studied till now: The founders of Calculus are Sir Isaac Newton and Gottfried Leibniz who worked on two different lines(separately) & founded the concepts of 'Differential' & 'Integral' Calculus. Leibniz thought in terms of infinitesimals while Newton typically thought in terms of limits though he also used infinitesimals. Both Newton and Leibniz gave proofs of their work in terms of geometry. This was the style of proof for their times which had been inherited from the Ancient Greeks. These proofs were did not work. The concept of a limit was given a rigorous definition in the 1750s and rigorous proofs of calculus where supplied from the early 18th century onward firstly by the french mathematician Cauchy. Infinitesimals were seen as intuitive but defective. In the late 1950s the american mathematician Abraham Robinson used concepts of mathematical logic to rigorously define infinitesimals. The ideas behind the calculus were extremely difficult to develop and the calculus had a rich history prior to Newton and Leibniz. The two branches of calculus provide answers to two different questions. Integral calculus asks what is the area under a curve? Differential calculus asks what is the slope of a curve? There were answers to these questions in more elementary branches of maths but the techniques were hard to develop and not general. The calculus is a set of tools to answer these questions in a more specific and general way. The two became one subject when Isaac Barrow discovered what is now known as the fundamental theorem of calculus. A simple first description is that the operations of finding integrals (that is areas) and differentials (that is slopes) are inverses of each other. = Newton worked on tangents & normals to curves & named Calculus 'The Theory Of Fluxions' whilst Leibniz worked on 'Areas Under Curves'. Calculus relies on the fundamental principle that the tangent to a curve touches the curve along an infinitesimal line and not at a point. From this we can form infinitesimal right-angled triangles under curves adjacent to certain points. This leads to the basic equation of calculus: f' = {f(x+E) - f(x)} / E or f(x+E) = f(x) + Ef'(x) Where E (epsilon) is an infinitesimal value; that is a number less than any rational number. The process of working out a formula for f'(x) from a given function is known as differentiation. Note that for a symmetrical curve it is easy to define the tangent; it is the line which is perpendicular to the axis of symmetry where it touches the curve. Considering the triangle adjacent to zero on y=x^2, these two principles can be used to show that E^2 = 0. This approach applies to all smooth curves and so is known as smooth infinitesimal analysis. It can be used to derive all of the rules of calculus in a simple algebraic manner. One of these is the Fundamental Theorem, which states that the function that yields the area under a curve up to a given point is found by carrying out the reverse of differentiation (integration) on the function of the curve. Hence, integration is sometimes also referred to as '''Anti-Differentiation'.'' Dev has categorized Calculus into the following Sections, which we will be studying in detail in this Wiki: 1)Pre-Calculus 2)Differential Calculus & its Applications 3)Integral Calculus & its Applications 4)Differential Equations 5)Advanced Calculus =